Kepler's third law - semi-major axis

In Kepler's third law (T² = 4pi²/GM r³), the r is the semi-major axis.

But why?

Why not the semi-minor axis? Or the average radius throughout the eccentric orbit?


(I'd be happy with a mathematical answer if that's easier)
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InteractiveMindAsked:
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Arthur_WoodConnect With a Mentor Commented:
In addition, Kepler used the Semi-Major axis because that is the measured parameter that characterizes the orbit.  Semi-minor axis cannot be used to characterize the orbit, since it is dependent on the Eccentricity of the orbit.

Kepler's 3rd law has no dependency, whatsoever, of the eccentricity of the orbit.  A very eccentiric orbit will have the same period as a very circular orbit, if both have the same Semi-major axis.  

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GnarOlakConnect With a Mentor Commented:
Math class was a long time ago so I don't even pretend to understand a lot of this but http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion has a proof of Kepler's Third Law.  Hope it helps.
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GnarOlakCommented:
Clicking that link won't work.  EE truncated the link at the apostrophe.  Copy the entire URL into a browser.
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d-glitchConnect With a Mentor Commented:
A planet is at the ends of the semi-major axis when it is closest to the sun (max kinetic energy)
and when it is furthest from the sun (max potential energy).
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ozoConnect With a Mentor Commented:
if you imagine orbits with an eccentricity close to 1, it may be more intuitive that semi-major axis makes more sense there than semi-minor axis
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grg99Connect With a Mentor Commented:
becuz mathematically the semi-minor axis can go to zero, and division by zero leads to big trouble.

But the other axis can only get bigger, which is mathematically tenable.

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Arthur_WoodCommented:
for a mathematical proof of the Third law (heavy on the algebra - just a warning), see:

http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Proving_Kepler.27s_third_law

(becuase of the ' in the URL, you will nedd to COPY (Ctrl-C) and Paste (Ctrl-V) the URL to the Browser, you cannot simply clink on the link that shows in this answer)

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Arthur_WoodConnect With a Mentor Commented:
Kepler did not state the 3rd law as :

T² = 4pi²/GM r³

but rather as a relationship that states the the square of the Period is PROPORTIONAL to the cube of then Semi-major axis.  This was taken entirely from observations of the actual motions of the planets.  Remember that Kepler was trying to match the positions of the planets, in the sky, as observed very accurately by Tycho Brahe (Kepler was NOT an observational astronomer - he used the observations provided by others, principally Tycho Brahe in Denmark, about 20 years before Kepler joined him as an assistant.

The mathematical formulation came much later, with the derivation from Newton's Law of Gravity.
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InteractiveMindAuthor Commented:
thank you all
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